The nuclear matrix elements (NMEs) of the neutrinoless double-$\beta$ ($0\nu \beta \beta$) decay in the candidate nuclei relevant to the current and next-generation experimental searches are crucial for extracting beyond-standard-model parameters from the experimental half-life measurements. However, the recently recognized short-range operator with an unknown coupling, required by the renormalizability of the leading-order $nn\to ppee$ amplitudes within the framework of chiral effective field theory (EFT), has introduced significant uncertainties into the NMEs. In this Letter, the $0\nu \beta \beta$-decay NMEs of ${}^{48}\mathrm{Ca}, {}^{76}\mathrm{Ge}, {}^{82}\mathrm{Se}, {}^{100}\mathrm{Mo}$, and ${}^{136}\mathrm{Xe}$ are calculated using, for the first time, the decay operator derived from the relativistic chiral EFT and the relativistic many-body wave functions obtained from the state-of-the-art relativistic configuration-interaction density functional theory. The resulting NMEs are free from the uncertainties typically introduced by leading-order short-range operators, as the relativistic chiral decay operator naturally ensures the renormalizability of the $nn\to ppee$ amplitudes at leading order, obviating the need for a short-range term. The present NMEs are then compared with those obtained using the nonrelativistic chiral decay operators, which includes both long- and short-range terms, and the short-range uncertainties are analyzed comprehensively. This study significantly reduces a key source of uncertainty in the evaluation of $0\nu \beta \beta$-decay NMEs. The reported NMEs without contamination of the leading-order short-range uncertainties are important for assessing the sensitivity to the inverted neutrino-mass hierarchy in next-generation experiments.

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• Electroweak interactions in nuclear physics
• Neutrinoless double beta decay
• Nuclear many-body theory
• Nuclear structure & decays